C₃×C₃₇×#6908 #41004712

order	26973
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₃⁴×C₃₇ · Schur multiplier (H₂) = C₃⁶

Aliases

direct product

C₃×C₃₇×#6908

Subgroups

	subgroup	G/H
center	C₃×C₉×C₃₇	H(3,𝔽₃)
commutator, Frattini, cosocle	C₃²	C₃⁴×C₃₇
socle	C₃²×C₃₇	C₃×H(3,𝔽₃)
LCS	G ▷ C₃² ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃×C₉×C₃₇ ◁ C₃²×C₉×C₃₇ ◁ G

1064 conjugacy classes of subgroups; only the 476 normal subgroups are shown.

normal subgroups only

This group has 1064 conjugacy classes of subgroups — too many to draw. Only its 476 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 67
transitive degree	μ_t(G) = 2997

A minimal faithful permutation representation — 4 generators on 67 points:

(1,2,3)(31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67);(4,5,8)(6,10,15)(7,9,16)(11,20,14)(12,18,23)(13,17,24)(19,27,22)(21,25,29)(26,30,28);(5,9,11)(10,17,19)(18,25,26);(4,6,12,7,13,21,14,22,28)(5,10,18,9,17,25,11,19,26)(8,15,23,16,24,29,20,27,30)

Conjugacy classes & character table

5661 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(26973, 483);